Experimental Results of the Search for Unitals in the Projective Planes of Order 25 Stoicho D. Stoichev Department of Computer Systems, Technical University of Sofia email:
[email protected] Abstract In this paper we present the results from a program developed by the author that finds the unitals of the known 193 projective planes of order 25. There are several planes for which we have not found any unital. One or more than one unitals have been found for most of the planes. The found unitals for a given plane are nonisomorphic each other. There are a few unitals isomorphic to a unital of another plane. A t - (v; k; λ) design D is a set X of points together with a family B of k-subsets of X called blocks with the property that every t points are contained in exactly λ blocks. The design with t = 2 is called a block-design. The block-design is symmetric if the role of the points and blocks can be changed and the resulting confguration is still a block-design. A projective plane of order n is a symmetric 2-design with v = n2 + n + 1, k = n + 1, λ = 1. The blocks of such a design are called lines. A unital in a projective plane of order n = q2 is a set U of q3 + 1 points that meet every line in one or q + 1 points. In the case projective planes of order n = 25 we have: q = 5 , the projective plane is 2 - (651; 26; 1) design, the unital is a subset of q 3 + 1 = 53+ 1 = 126 points and every line meets 1 or 6 points from the subset Key words: projective plane, design, graph, isomorphism, automorphism, group, stabilizer, exact algorithm, heuristic algorithm, partition, generators, orbits and order of the graph automorphism group.
Article Outline 1. Introduction 2. Experimental results 3. Concluding remarks Acknowledgements References
1. Introduction We assume familiarity with the basics of the combinatorial designs (cf., e.g. [1]). A t - (v; k; λ) design D [1] is a set X of points together with a family B of k-subsets of X called blocks with the property that every t points are contained in exactly λ blocks. The design with t = 2 is called a block-design. The block-design is symmetric if the role of the points and blocks can be changed and the resulting confguration is still a block-design. A projective plane of order n is a symmetric 2-design with v = n2 + n + 1, k = n + 1, λ = 1. The blocks of such a design are called lines. A unital in a projective plane of order n = q2 is a set U of q3 + 1 points that meet every line in one or q + 1 points. In the case of projective planes of order n = 16 we have: q = 4 , the projective plane is 2 (273; 17; 1) design, the unital is a subset of q 3 + 1 = 43+ 1 = 65 points and every line meets 1 or 5 points from the subset. In the case of projective planes of order n = 25 we have: q = 5 , the projective plane is 2 (651; 26; 1) design, the unital is a subset of q 3 + 1 = 53+ 1 = 126 points and every line meets 1 or 6 points from the subset.
The experimental results of the algorithm in [5] for the known planes of order 16 [4] are given in [6].
2. Experimental results In [8] we present the text of the current paper and the experimental results (in Appendix) from a program that finds the unitals in all 193 known projective planes of order 25 from the website [3].. The vertex labels in this websites start from 0, but in our program and results the starting label is 1..The program we use in present paper is based on the algorithm described in [5]:. In [5] we describe the algorithm for finding unitals and maximal arcs in projective planes of order 16. In this paper we use the same algorithm but with parameters for a plane of order 25. The algorithm is heuristic – it does not guarantee finding all possible unitals of a given plane. All fond unitals of an unital are nonisomorphic each other. There are planes with no found unitals. The results for each plane contain values for the following variables: 1. Name of the plane 2. Order of the plane automorphism group 3. Number of the orbits of the plane automorphism gtoup 4. Order of the unital automorphism group 5. Number of the orbits of the unital automorphism group 6. Sizes of the orbits of the unital automorphism group 7. Labels of the vertices of the unital The information for 1 to 3 is present for each plane and the information for 4 to 7 is present for each found unital. Example (for the plane A1.HTM, see the text below - from the line with ‘A1.HTM’ : The order of the plane automorphism group is 360000 and the number of its orbits is 8. Then, the results for all 9 found unitals follow. The orders of their automorphism groups are 144, 24, 20, 20, 15, 10, 10, 10, 10. A1.HTM ORDER OF THE PLANE AUTOMORPHISM GROUP 360000 NUMBER OF THE ORBITS OF THE PLANE AUTOMORPHISM GTOUP= 8 ORDER OF THE UNITAL AUTOMORPHISM GROUP= 144 NUMBER OF THE ORBITS OF THE UNITAL AUTOMORPHISM GROUP = SIZES OF THE ORBITS OF THE UNITAL AUTOMORPHISM GROUP= 1- 72 2- 48 36 UNITAL= 6 7 8 9 10 11 30 31 32 34 35 39 181 182 184 185 186 187 188 189 190 191 192 193 194 196 197 198 199 209 210 211 212 213 214 215 216 229 232 237 238 239 240 241 243 244 245 246 247 248 249 250 251 252 266 270 271 274 275 277 278 279 289 290 291 292 293 294 295 296 297 298 299 300 303 304 307 308 309 353 355 357 359 361 363 364 365 366 367 369 376 378 379 381 384 387 606 607 608 609 611 615 616 617 619 620 621 628 632 634 635 636 638 640 643 646 647 648 651 ORDER OF THE UNITAL AUTOMORPHISM GROUP= 24 NUMBER OF THE ORBITS OF THE UNITAL AUTOMORPHISM GROUP = SIZES OF THE ORBITS OF THE UNITAL AUTOMORPHISM GROUP= 1- 24 2- 24 3- 24 4- 24 5- 24 64 71 81 UNITAL=
3
183 208 242 276 301 368 618
8
1 68
10
11
22
23
25
40
41
42
49
50
51
64
65
70 71 85 91 94 95 97 98 118 119 120 125 127 133 138 142 148 149 154 173 174 176 177 178 179 183 184 186 190 196 197 200 202 205 206 207 210 214 216 223 225 226 227 229 233 238 239 245 249 251 254 259 262 264 267 271 273 275 278 329 331 336 340 345 389 392 397 398 406 409 410 411 412 413 423 431 435 442 446 451 453 460 462 480 481 489 492 538 541 553 570 573 576 577 593 595 600 603 608 611 619 620 635 636 ORDER OF THE UNITAL AUTOMORPHISM GROUP= 20 NUMBER OF THE ORBITS OF THE UNITAL AUTOMORPHISM GROUP = SIZES OF THE ORBITS OF THE UNITAL AUTOMORPHISM GROUP= 1- 20 2- 20 3- 20 4- 20 5- 20 6- 20 75 81 UNITAL= 8 27 29 30 32 48 49 53 61 62 66 71 79 82 104 110 115 121 129 133 138 143 149 159 162 168 173 195 197 204 206 207 220 225 229 237 239 241 243 245 248 257 263 264 276 281 286 289 293 297 306 315 319 325 327 329 332 334 337 345 354 358 367 370 377 385 392 396 405 409 411 415 419 423 431 437 446 450 453 456 458 470 472 473 477 478 489 490 495 505 515 518 519 523 524 531 533 536 537 544 554 557 559 560 568 571 573 587 595 602 603 604 605 609 619 620 626 640 641 ORDER OF THE UNITAL AUTOMORPHISM GROUP= 20 NUMBER OF THE ORBITS OF THE UNITAL AUTOMORPHISM GROUP = SIZES OF THE ORBITS OF THE UNITAL AUTOMORPHISM GROUP= 1- 20 2- 20 3- 20 4- 20 5- 20 6- 20 75 81 UNITAL= 8 27 32 33 35 41 51 53 55 61 71 72 74 89 97 98 101 104 111 117 119 121 133 140 158 167 168 179 195 206 208 216 217 220 225 230 236 241 245 254 256 259 260 266 276 280 282 290 293 295 297 302 314 315 329 330 339 340 341 350 360 362 366 368 371 375 377 378 401 405 410 411 414 421 439 440 446 449 456 458 460 476 478 482 486 488 492 498 512 523 527 531 533 534 536 537 540 542 544 549 551 554 555 557 567 568 573 574 603 604 613 621 624 629 630 634 637 643 648 ORDER OF THE UNITAL AUTOMORPHISM GROUP= 15 NUMBER OF THE ORBITS OF THE UNITAL AUTOMORPHISM GROUP = SIZES OF THE ORBITS OF THE UNITAL AUTOMORPHISM GROUP= 1- 15 2- 15 3- 15 4- 15 5- 15 6- 15 7- 15 8- 15 95 101 UNITAL=
67 134 195 231 326 420 543
8
101 202 266 339 424 498 567
8
91 205 270 347 425 514 560
10
24 88
28
36
45
48
49
51
56
57
66
67
68
75
79
86
98 107 117 124 125 127 134 135 139 140 143 146 150 152 157 160 161 163 164 173 177 179 195 200 202 204 206 212 215 232 242 244 245 270 275 280 283 288 293 295 296 297 303 309 315 318 319 323 326 327 332 333 348 361 364 365 371 372 373 379 380 384 385 389 391 403 406 411 416 420 427 435 437 438 441 444 448 451 454 464 466 470 475 482 485 488 491 495 537 555 556 562 564 581 598 599 605 609 615 618 622 623 626 627 635 636 643 644 647 650 ORDER OF THE UNITAL AUTOMORPHISM GROUP= 10 NUMBER OF THE ORBITS OF THE UNITAL AUTOMORPHISM GROUP = 14 SIZES OF THE ORBITS OF THE UNITAL AUTOMORPHISM GROUP= 1- 10 2- 10 3- 10 4- 10 5- 10 6- 10 7- 10 8- 10 9- 10 10- 10 1110 12- 10 135 141 UNITAL= 10 27 31 37 39 43 46 54 56 61 64 67 72 78 85 88 95 100 114 116 118 131 133 147 148 153 155 163 174 182 190 201 209 212 214 216 217 218 226 234 244 246 254 256 258 260 263 264 283 285 287 294 302 303 306 310 315 320 326 327 343 346 349 350 356 359 363 364 370 373 374 376 394 398 401 411 413 418 424 425 433 443 448 452 458 463 465 467 470 475 479 495 504 511 516 518 524 530 536 539 541 544 546 549 553 555 564 565 568 570 572 593 595 597 601 603 606 608 610 622 629 630 634 639 645 648 ORDER OF THE UNITAL AUTOMORPHISM GROUP= 10 NUMBER OF THE ORBITS OF THE UNITAL AUTOMORPHISM GROUP = 14 SIZES OF THE ORBITS OF THE UNITAL AUTOMORPHISM GROUP= 1- 10 2- 10 3- 10 4- 10 5- 10 6- 10 7- 10 8- 10 9- 10 10- 10 1110 12- 10 135 141 UNITAL= 10 27 30 35 36 41 48 57 58 59 69 75 78 81 83 85 87 91 99 103 106 114 117 131 138 145 148 153 158 161 162 171 176 184 188 191 192 197 200 211 216 220 222 223 230 233 251 254 259 268 269 276 278 283 284 291 304 307 309 311 319 322 325 341 359 361 366 379 380 384 388 396 398 403 409 421 422 423 429 441 442 447 449 450 451 457 469 473 474 476 485 488 491 493 499 509 510 512 513 520 528 530 535 539 544 550 551 564 568 574 576 580 581 584 594 595 597 603 609 613 614 615 620 627 642 643 ORDER OF THE UNITAL AUTOMORPHISM GROUP= 10 NUMBER OF THE ORBITS OF THE UNITAL AUTOMORPHISM GROUP = 14
SIZES OF THE ORBITS OF THE UNITAL AUTOMORPHISM GROUP= 1- 10 2- 10 3- 10 4- 10 5- 10 6- 10 7- 10 8- 10 9- 10 10- 10 1110 12- 10 135 141 UNITAL= 10 27 55 56 60 64 74 78 82 85 89 94 95 98 100 108 114 117 120 126 131 132 133 135 143 148 151 153 161 174 175 179 180 193 201 205 206 207 208 215 216 223 226 234 235 238 243 245 249 254 255 259 261 263 266 267 283 285 292 294 304 307 308 310 314 318 320 328 331 334 335 340 344 346 356 358 359 370 377 378 388 394 395 397 398 400 432 435 437 439 455 458 467 479 481 482 491 493 501 505 506 509 521 530 539 544 548 556 561 564 568 585 588 595 596 597 598 603 606 610 622 632 638 639 647 649 ORDER OF THE UNITAL AUTOMORPHISM GROUP= 10 NUMBER OF THE ORBITS OF THE UNITAL AUTOMORPHISM GROUP = 14 SIZES OF THE ORBITS OF THE UNITAL AUTOMORPHISM GROUP= 1- 10 2- 10 3- 10 4- 10 5- 10 6- 10 7- 10 8- 10 9- 10 10- 10 1110 12- 10 135 141 UNITAL= 10 27 28 30 35 41 45 56 62 64 71 78 84 85 87 95 100 106 107 114 119 124 131 133 138 148 153 166 167 174 177 181 186 189 191 194 201 216 220 225 226 231 234 251 254 263 268 271 274 280 281 283 285 289 294 301 310 317 319 320 324 341 342 346 354 355 356 359 361 366 370 380 389 394 398 403 405 409 423 445 450 457 458 460 466 467 472 476 479 484 485 492 499 500 514 515 519 530 531 535 537 538 539 544 550 564 568 576 582 584 595 597 600 602 603 606 610 613 616 622 626 627 639 641 642 650
3. Concluding remarks By our algorithm we have found new unitals in projective plane of order 25, but not all of them. The following approaches can be used to find more or all unitals : (a) Development of improved algorithms by finding new conditions for pruning the search tree; (b) Transformation of the solution for one plane to solution for another plane (R. Mathon's approach - in private communication); (c) Development of parallel algorithms.
Acknowledgements The author would like to thank Vladimir Tonchev for suggesting the problem of developing an algorithm for finding unitals in projective planes and for giving the general idea of such an algorithm - use of unions of orbits, and for extensive discussions and exchanges for many years. .
References [1] Colbourn C. J., J. H. Dinitz (Eds.) The CRC handbook of Combinatorial Designs, CRC Press, New York, 1996. [2] Hamilton N., S. D. Stoichev , V. D. Tonchev. Maximal Arcs and Disjoint Maximal Arcs in Projective Planes of Order 16. J. Geom. 67 (2000), 117{126. [3] Projective Planes of Order 25: http://www.cs.uwa.edu/au/~gordon/planes16/index#planes [4] Royle G. F. Known planes of Order 16: http://www.cs.uwa.edu/au/~gordon/planes16/index#planes [5] Stoichev S. D. Algorithms for finding unitals and maximal arcs in projective planes of order 16, Serdica J. Computing 1 (2007), 279-292. [6] Stoichev S. D., V. D. Tonchev. Unital Designs in Planes of Order 16. Discrete Applied Mathematics 102 (2000), 151{158. [7] Stoichev S. D. Polynomial time and space exact and heuristic algorithms for determining the generators, orbits and order of the graph automorphism group, arXiv:1007.1726v2 [cs.DS] (2010). [8] Stoichev S. D. Experimental Results of the Search for Unitals in Projective Planes of Order 25,
http://sdstoichev1.wordpress.com/2012/02/16/experimental-results-of-the-search-for-unitalsin-projective-planes-of-order-25/
